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I trained in a style of karate which strictly regulated breathing and found it to be very effective. I am sure most readers with experience in breathing techniques will have a lot of technical advice regarding this.
In my studies of aikido I was advised to simply relax and the breathing will take care of itself. Surprisingly I found this to be the best advice. The key being to relax. Keep in mind the ability to relax during intense physical or psychological stress is not easy. But, if you can manage it you will find the breathing comes naturally. I know this advice seems simplistic but I think it is worth studying.

As I have learned, even more so in the last few months, there is alot more to breathing than just the basics we have discussed here. If you follow the internal martial arts threads here.... a great deal of that discussion centers around the connection between movement, breathing, the environment around you.

all those exercises we do in aikido (or are supposed to do), are designed to help us do this better.

While the basics are covered above....I think it is a life long study getting it correct.

That's precisely how I see it, too. And the breathing has to be done with your center.

Another thing with breathing: You will improve significantly, if you start your breathing before you start moving your body. For example, start exhaling before you strike or throw or pin.
The breathing should feel like it is what actually performs the technique, while the arms merely follow.

this might seem a little facetious, but...
Breathe In - contract the diaphragm and intercostal muscles, increasing the size of the chest cavity. This causes a decrease in the pressure in the very thin layer of fluid between the rib cage and the lungs, which causes the lungs to expand. Lungs expanding causes a decrease in air pressure within the lungs, which causes air to rush from the higher pressure (outside the person) to the lower pressure (inside the person).
Breathe Out - reverse of the above.

The body has some very sophisticated mechanisms that respond to movement, carbon dioxide in the blood, and a whole lot of other things to regulate breathing - the trick is letting the body do its job.

Interesting. This is something I'm working on. Although, I have no idea if we're talking about it in the same way. I'm using the breath (the inhale) as a tool to help inititiate internal pressure, that powers the raising of the arms. And I'm not talking about the oxygen. The exhale and gravity lowers the arms (kokyu anyone?). These are not overt cycles. My goal is a very calm or imperceptible breath, and to always use the legs.

An early sensei of mine taught me to inhale quickly and deeply through the nose as uke attacks (breath in uke) and exhale slower and evenly through the mouth during the execution of the technique (exhale uke).
This helps perform the technique smoothly and powerfully.